Physica D: Nonlinear Phenomena [5 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0167-2789 Published by Elsevier [2563 journals] [SJR: 0.976] [H-I: 83] |
- Quasiperiodicity and phase locking in stochastic circle maps: A spectral
approach- Abstract: Publication date: Available online 14 August 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Alla Borisyuk , Firas Rassoul-Agha
While there are clear definitions of what it means for a deterministic dynamical system to be periodic, quasiperiodic, or chaotic, it is unclear how to define such notions for a noisy system. In this article, we study Markov chains on the circle, which is a natural stochastic analogue of deterministic dynamical systems. The main tool is spectral analysis of the transition operator of the Markov chain. We analyze path-wise dynamic properties of the Markov chain, such as stochastic periodicity (or phase locking) and stochastic quasiperiodicity, and show how these properties are read off of the geometry of the spectrum of the transition operator.
PubDate: 2014-08-16T17:21:01Z
- Abstract: Publication date: Available online 14 August 2014
- Impact of network connectivity on the synchronization and global dynamics
of coupled systems of differential equations- Abstract: Publication date: Available online 12 August 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Peng Du , Michael Y. Li
The global dynamics of coupled systems of differential equations defined on an interaction network are investigated. Local dynamics at each vertex, when interactions are absent, are assumed to be simple: solutions to each vertex system are assumed to converge to an equilibrium, either on the boundary or in the interior of the feasible region. The interest is to investigate the collective behaviours of the coupled system when interactions among vertex systems are present. It was shown in Li and Shuai (2010) that, if the interaction network is strongly connected, then solutions to the coupled system synchronize at a single equilibrium. We focus on the case when the underlying network is not strongly connected and the coupled system may have mixed equilibria whose coordinates are in the interior at some vertices while on the boundary at others. We show that solutions on a strongly connected component of the network will synchronize. Considering a condensed digraph by collapsing each strongly connected component, we are able to introduce a partial order on the set P of all equilibria, and show that all solutions of the coupled system converge to a unique equilibrium P ∗ that is the maximizer in P . We further establish that behaviours of the coupled system at minimal elements of the condensed digraph determine whether the global limit P ∗ is a mixed equilibrium. The theory are applied to mathematical models from epidemiology and spatial ecology.
PubDate: 2014-08-16T17:21:01Z
- Abstract: Publication date: Available online 12 August 2014
- Optimal entrainment with smooth, pulse, and square signals in weakly
forced nonlinear oscillators- Abstract: Publication date: Available online 14 August 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Hisa-Aki Tanaka
A physical limit of entrainability of nonlinear oscillators is considered for an external weak signal (forcing). This limit of entrainability is characterized by the optimization problem maximizing the width of the Arnold tongue (the frequency-locking range vs forcing magnitude) under certain practical constraints. Here we show a solution to this optimization problem, thanks to a direct link to Hölder’s inequality. This solution defines an ideal forcing realizing the entrainment limit, and as the result, a fundamental limit of entrainment is clarified as follows. For 1 : 1 entrainment, we obtain (i) a construction of the global optimal forcing and a condition for its uniqueness in L p -space with p > 1 , and (ii) a construction of the global optimal pulse-like forcings in L 1 -space, and for m : n entrainment ( m ≠ n ), some informations about the non-existence of the ideal forcing. (iii) In addition, we establish definite algorithms for obtaining the global optimal forcings for 1 < p ≤ ∞ and these pulse-like forcings for p = 1 . These theoretical findings are verified by systematic, extensive numerical calculations and simulations.
PubDate: 2014-08-16T17:21:01Z
- Abstract: Publication date: Available online 14 August 2014
- From synchronisation to persistent optical turbulence in laser arrays
- Abstract: Publication date: Available online 4 August 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Nicholas Blackbeard , Sebastian Wieczorek , Hartmut Erzgräber , Partha Sharathi Dutta
We define and study synchronisation in a linear array of nearest-neighbour coupled lasers. Our focus is on possible synchronisation types and the stability of their corresponding synchronisation manifolds with dependence on the coupling strength, the laser frequency detuning, the amount of shear (amplitude-phase coupling) in a single laser, and the array size. We classify, and give analytical conditions for the existence of complete synchronisation solutions, where all the lasers emit light with the same intensity and frequency. Furthermore, we derive stability criteria for two special cases where all the lasers oscillate (i) in-phase with each other and (ii) in anti-phase with their nearest neighbour(s). We then explain transitions from complete synchronisation, to partial synchronisation (where only a subset of the lasers synchronise), to persistent optical turbulence (where no lasers synchronise and each laser is chaotic) in terms of bifurcations including blowouts of chaotic attractors. Finally, we quantify properties of optical turbulence using Lyapunov spectrum and dimension, which highlights differences in chaos generated by nearest-neighbour and globally coupled oscillators.
PubDate: 2014-08-06T17:17:20Z
- Abstract: Publication date: Available online 4 August 2014
- Quasi-periodic solutions for quasi-periodically forced nonlinear
Schrödinger equations with quasi-periodic inhomogeneous terms- Abstract: Publication date: Available online 1 August 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Jie Rui , Jianguo Si
In this paper, it is shown that there exist many small amplitude quasi-periodic solutions for non-autonomous, quasi-periodically forced in time nonlinear Schrödinger equations with quasi-periodic inhomogeneous terms, under periodic spatial boundary conditions, via KAM theory.
PubDate: 2014-08-02T17:16:38Z
- Abstract: Publication date: Available online 1 August 2014
- Analysis of a temperature-dependent model for adhesive contact with
friction- Abstract: Publication date: Available online 1 July 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Elena Bonetti , Giovanna Bonfanti , Riccarda Rossi
We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid support, encompassing thermal and frictional effects. Following Frémond’s approach, adhesion is described in terms of a surface damage parameter χ . The related equations are the (quasistatic) momentum balance for the vector of displacements, and parabolic-type evolution equations for χ , and for the absolute temperatures of the body and of the adhesive substance on the contact surface. All of the constraints on the internal variables, as well as the contact and the friction conditions, are rendered by means of subdifferential operators. Furthermore, the temperature equations, derived from an entropy balance law, feature singular functions. Therefore, the resulting PDE system has a highly nonlinear character. After introducing a suitable regularization of the Coulomb law for dry friction, we address the analysis of the resulting PDE system. The main result of the paper states the existence of global-in-time solutions to the associated Cauchy problem. It is proved by passing to the limit in a carefully tailored approximate problem, via variational techniques.
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: Available online 1 July 2014
- The inviscid, compressible and rotational, 2D isotropic Burgers and
pressureless Euler–Coriolis fluids; solvable models with
illustrations- Abstract: Publication date: Available online 15 July 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Ph. Choquard , M. Vuffray
The coupling between dilatation and vorticity, two coexisting and fundamental processes in fluid dynamics Wu et al. (2006, pp. 3, 6) is investigated here, in the simplest cases of inviscid 2D isotropic Burgers and pressureless Euler–Coriolis fluids respectively modeled by single vortices confined in compressible, local, inertial and global, rotating, environments. The field equations are established, inductively, starting from the equations of the characteristics solved with an initial Helmholtz decomposition of the velocity fields namely a vorticity free and a divergence free part Wu et al. (2006, Sects. 2.3.2, 2.3.3) and, deductively, by means of a canonical Hamiltonian Clebsch like formalism Clebsch (1857, 1859), implying two pairs of conjugate variables. Two vector valued fields are constants of the motion: the velocity field in the Burgers case and the momentum field per unit mass in the Euler–Coriolis one. Taking advantage of this property, a class of solutions for the mass densities of the fluids is given by the Jacobian of their sum with respect to the actual coordinates. Implementation of the isotropy hypothesis entails a radial dependence of the velocity potentials and of the stream functions associated to the compressible and to the rotational part of the fluids and results in the cancellation of the dilatation-rotational cross terms in the Jacobian. A simple expression is obtained for all the radially symmetric Jacobians occurring in the theory. Representative examples of regular and singular solutions are shown and the competition between dilatation and vorticity is illustrated. Inspired by thermodynamical, mean field theoretical analogies, a genuine variational formula is proposed which yields unique measure solutions for the radially symmetric fluid densities investigated. We stress that this variational formula, unlike the Hopf-Lax formula, enables us to treat systems which are both compressible and rotational. Moreover in the one-dimensional case, we show for an interesting application that both variational formulas are equivalent.
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: Available online 15 July 2014
- Binocular rivalry waves in a directionally selective neural field model
- Abstract: Publication date: Available online 16 July 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Samuel R. Carroll , Paul C. Bressloff
We extend a neural field model of binocular rivalry waves in the visual cortex to incorporate direction selectivity of moving stimuli. For each eye, we consider a one-dimensional network of neurons that respond maximally to a fixed orientation and speed of a grating stimulus. Recurrent connections within each one-dimensional network are taken to be excitatory and asymmetric, where the asymmetry captures the direction and speed of the moving stimuli. Connections between the two networks are taken to be inhibitory (cross-inhibition). As per previous studies, we incorporate slow adaption as a symmetry breaking mechanism that allows waves to propagate. We derive an analytical expression for traveling wave solutions of the neural field equations, as well as an implicit equation for the wave speed as a function of neurophysiological parameters, and analyze their stability. Most importantly, we show that propagation of traveling waves are faster in the direction of stimulus motion than against it, which is in agreement with previous experimental and computational studies.
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: Available online 16 July 2014
- Transient behavior of collapsing ring solutions in the critical nonlinear
Schrödinger equation- Abstract: Publication date: 15 September 2014
Source:Physica D: Nonlinear Phenomena, Volume 284
Author(s): Jordan Allen-Flowers , Karl B. Glasner
The critical nonlinear Schrödinger equation (NLS) possesses nearly self-similar ring profile solutions. We address the question of whether this profile is maintained all the way until the point of singularity. A perturbative analysis of the rescaled PDE and the resulting self-similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak-shaped collapse instead. A numerical scheme capable of resolving self-similar behavior to high resolutions confirms our analysis. We also consider ring-type blowup arising either from azimuthally polarized solutions of a coupled NLS system or as vortex solutions of the usual NLS. In this case, the ring profile is maintained up to the time of singularity.
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: 15 September 2014
- Editorial Board
- Abstract: Publication date: 15 August 2014
Source:Physica D: Nonlinear Phenomena, Volume 283
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: 15 August 2014
- Nonautonomous analysis of steady Korteweg–de Vries waves under
nonlocalised forcing- Abstract: Publication date: Available online 17 July 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Sanjeeva Balasuriya , Benjamin J. Binder
Recently developed nonautonomous dynamical systems theory is applied to quantify the effect of bottom topography variation on steady surface waves governed by the Korteweg–de Vries (KdV) equation. Arbitrary (but small) nonlocalised bottom topographies are amenable to this method. Two classes of free surface solutions—hyperbolic and homoclinic solutions of the associated augmented dynamical system—are characterised. The first of these corresponds to near-uniform free-surface flows, and explicit formulæ are developed for a range of topographies. The second corresponds to solitary waves on the free surface, and a method for determining their number is developed. Formulæ for the shape of these solitary waves are also obtained. Theoretical free-surface profiles are verified using numerical KdV solutions, and excellent agreement is obtained.
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: Available online 17 July 2014
- Spatial structure of Sinai–Ruelle–Bowen measures
- Abstract: Publication date: 1 October 2014
Source:Physica D: Nonlinear Phenomena, Volume 285
Author(s): N. Chernov , A. Korepanov
Sinai–Ruelle–Bowen measures are the only physically observable invariant measures for billiard dynamical systems under small perturbations. These measures are singular, but as it was noted in Bonetto et al. (2012), marginal distributions of spatial and angular coordinates are absolutely continuous. We generalize these facts and provide full mathematical proofs.
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: 1 October 2014
- Homoclinic complexity in the localised buckling of an extensible
conducting rod in a uniform magnetic field- Abstract: Publication date: 15 September 2014
Source:Physica D: Nonlinear Phenomena, Volume 284
Author(s): Caifa Guo , G.H.M. van der Heijden , Hong Cai
We study the localised buckling of an extensible conducting rod subjected to end loads and placed in a uniform magnetic field. The trivial straight but twisted rod is described by a fixed point of a four-dimensional Hamiltonian system of equations previously shown to be chaotic. Localised solutions are given by homoclinic orbits to this fixed point and we explore the spatial complexity of localised rod configurations by means of shooting and parameter continuation methods that exploit the reversibility of the system of equations. Unlike in localisation studies of non-magnetic rods we find that for certain parameter values multiple Hamiltonian–Hopf bifurcations occur. Where these collide as parameters are varied, solutions exhibit delocalisation–relocalisation behaviour. Our results predict buckling instabilities and post-buckling behaviour of rods under combined mechanical and magnetic loads, which are relevant for electrodynamic space tethers and potentially for conducting nanowires in future electromechanical devices.
PubDate: 2014-07-28T17:15:45Z
- Abstract: Publication date: 15 September 2014
- Growth-induced blisters in a circular tube
- Abstract: Publication date: Available online 5 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): R. De Pascalis , G. Napoli , S.S. Turzi
The growth of an elastic film adhered to a confining substrate might lead to the formation of delimitation blisters. Many results have been derived when the substrate is flat. The equilibrium shapes, beyond small deformations, are determined by the interplay between the sheet elastic energy and the adhesion potential due to capillarity. Here, we study a non-trivial generalization to this problem and consider the adhesion of a growing elastic loop to a confining circular substrate. The fundamental equations, i.e., the Euler Elastica equation, the boundary conditions and the transversality condition, are derived from a variational procedure. In contrast to the planar case, the curvature of the delimiting wall appears in the transversality condition, thus acting as a further source of adhesion. We provide the analytic solution to the problem under study in terms of elliptic integrals and perform the numerical and the asymptotic analysis of the characteristic lengths of the blister. Finally, and in contrast to previous studies, we also discuss the mechanics and the internal stresses in the case of vanishing adhesion. Specifically, we give a theoretical explanation to the observed divergence of the mean pressure exerted by the strip on the container in the limit of small excess-length.
Graphical abstract
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 5 June 2014
- Quantifying force networks in particulate systems
- Abstract: Publication date: Available online 9 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Miroslav Kramár , Arnaud Goullet , Lou Kondic , Konstantin Mischaikow
We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes of these distributions: digital, position, and interaction, motivated by different types of data that may be available from experiments and simulations, e.g. digital images, location of the particles, and the forces between particles, respectively. We describe how algebraic topology, in particular, homology allows one to obtain algebraic representations of the geometry captured by these complexes. For each complex we define an associated force network from which persistent homology is computed. Using numerical data obtained from discrete element simulations of a system of particles undergoing slow compression, we demonstrate how persistent homology can be used to compare the force distributions in different systems, and discuss the differences between the properties of digital, position, and interaction force networks. To conclude, we formulate well-defined measures quantifying differences between force networks corresponding to different states of a system, and therefore allow to analyze in precise terms dynamical properties of force networks.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 9 June 2014
- Nonlinear chains inside walls
- Abstract: Publication date: Available online 11 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): D. Hennig , C. Mulhern
The conservative dynamics of a 1D chain of units coupled with (FPU type) nonlinear interactions is considered. Stationary patterns in such chains emerge due to a balance of coupling energy between neighbouring units. Particularly interesting are the nontrivial stationary states which contain segments of positive and negative slope. This results in a zig-zag pattern, in the case of periodic boundary conditions, and in kink (or anti-kink) solutions in the case of the free boundary conditions. Imposing constraints on the chain, by way of two confining infinitely high walls, has repercussions for the stability of these stationary states. Here, such stationary states, commensurable with the available space between the two walls, are examined in detail, and their respective stability properties are determined analytically by invoking the transfer matrix method. Strikingly, stationary anti-kink solutions and periodic zig-zag states, being unstable in the absence of confining walls, become stable when confining walls are introduced. Furthermore, simulations reveal that chains with randomly generated initial conditions can seek out these patterns, thus localising energy, and persist for considerable time.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 11 June 2014
- Large fluctuations of the nonlinearities in isotropic turbulence.
Anisotropic filtering analysis- Abstract: Publication date: Available online 11 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): D. Tordella , S. Di Savino , L. Sitzia
Using a Navier–Stokes isotropic turbulent field numerically simulated in a box with a discretization of 10243 (Biferale et al., 2005), we show that the probability of having a stretching-tilting larger than a few times the local enstrophy is low. By using an anisotropic kind of filter in the Fourier space, where wavenumbers that have at least one component below a threshold or inside a range are removed, we analyze these survival statistics when the large, the small inertial or the small inertial and dissipation scales are filtered out. By considering a flow obtained by randomising the phases of the Fourier modes, and applying our filtering techniques, we identified clearly the properties attributable to turbulence. It can be observed that, in the unfiltered isotropic Navier–Stokes field, the probability of the ratio ( ω ⋅ ∇ U / ω 2 ) being higher than a given threshold is higher than in the fields where the large scales were filtered out. At the same time, it is lower than in the fields were the small inertial and dissipation range of scales is filtered out. This is basically due to the suppression of compact structures in the ranges that have been filtered in different ways. The partial removal of the background of filaments and sheets does not have a first order effect on these statistics. These results are discussed in the light of a hypothesized relation between vortical filaments, sheets and blobs in physical space and in Fourier space. The study in fact can be viewed as a kind of test for this idea and tries to highlight its limits. We conclude that a qualitative relation in physical space and in Fourier space can be supposed to exist for blobs only. That is for the near isotropic structures which are sufficiently described by a single spatial scale and do not suffer from the disambiguation problem as filaments and sheets do. Information is also given on the filtering effect on statistics concerning the inclination of the strain rate tensor eigenvectors with respect to vorticity. In all filtered ranges, eigenvector 2 reduces its alignment, while eigenvector 3 reduces its misalignment. All filters increase the gap between the most extensional eigenvalue 〈 λ 1 〉 and the intermediate one 〈 λ 2 〉 and the gap between this last 〈 λ 2 〉 and the contractile eigenvalue 〈 λ 3 〉 . When the large scales are missing, the modulus of the eigenvalue 1 becomes nearly equal to that of the eigenvalue 3, similarly to the modulus of the associated components of the enstrophy production.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 11 June 2014
- Energy and potential enstrophy flux constraints in quasi-geostrophic
models- Abstract: Publication date: Available online 13 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Eleftherios Gkioulekas
We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-layer quasi-geostrophic models. Its physical significance is that it can diagnose whether any given multi-layer model that allows co-existing downscale cascades of energy and potential enstrophy can allow the downscale energy flux to become large enough to yield a mixed energy spectrum where the dominant k − 3 scaling is overtaken by a subdominant k − 5 / 3 contribution beyond a transition wavenumber k t situated in the inertial range. The validity of the flux inequality implies that this scaling transition cannot occur within the inertial range, whereas a violation of the flux inequality beyond some wavenumber k t implies the existence of a scaling transition near that wavenumber. This flux inequality holds unconditionally in two-dimensional Navier–Stokes turbulence, however, it is far from obvious that it continues to hold in multi-layer quasi-geostrophic models, because the dissipation rate spectra for energy and potential enstrophy no longer relate in a trivial way, as in two-dimensional Navier–Stokes. We derive the general form of the energy and potential enstrophy dissipation rate spectra for a generalized symmetrically coupled multi-layer model. From this result, we prove that in a symmetrically coupled multi-layer quasi-geostrophic model, where the dissipation terms for each layer consist of the same Fourier-diagonal linear operator applied on the streamfunction field of only the same layer, the flux inequality continues to hold. It follows that a necessary condition to violate the flux inequality is the use of asymmetric dissipation where different operators are used on different layers. We explore dissipation asymmetry further in the context of a two-layer quasi-geostrophic model and derive upper bounds on the asymmetry that will allow the flux inequality to continue to hold. Asymmetry is introduced both via an extrapolated Ekman term, based on a 1980 model by Salmon, and via differential small-scale dissipation. The results given are mathematically rigorous and require no phenomenological assumptions about the inertial range. Sufficient conditions for violating the flux inequality, on the other hand, require phenomenological hypotheses, and will be explored in future work.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 13 June 2014
- A mechanical counterexample to KAM theory with low regularity
- Abstract: Publication date: Available online 13 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Stefano Marò
We give a mechanical example concerning the fact that some regularity is necessary in KAM theory. We consider the model given by the vertical bouncing motion of a ball on a periodically moving plate. Denoting with f the motion of the plate, some variants of Moser invariant curve theorem apply if f ̇ is small in norm C 5 and every motion has bounded velocity. This is not possible if the function f is only C 1 . Indeed we construct a function f ∈ C 1 with arbitrary small derivative in norm C 0 for which a motion with unbounded velocity exists.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 13 June 2014
- Editorial Board
- Abstract: Publication date: 15 June 2014
Source:Physica D: Nonlinear Phenomena, Volumes 278–279
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 15 June 2014
- Optical beam shaping and diffraction free waves: A variational approach
- Abstract: Publication date: Available online 14 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): John A. Gemmer , Shankar C. Venkataramani , Charles G. Durfee , Jerome V. Moloney
We investigate the problem of shaping radially symmetric annular beams into desired intensity patterns along the optical axis. Within the Fresnel approximation, we show that this problem can be expressed in a variational form equivalent to the one arising in phase retrieval. Using the uncertainty principle we prove various rigorous lower bounds on the functional; these lower bounds estimate the L 2 error for the beam shaping problem in terms of the design parameters. We also use the method of stationary phase to construct a natural ansatz for a minimizer in the short wavelength limit. We illustrate the implications of our results by applying the method of stationary phase coupled with the Gerchberg-Saxton algorithm to beam shaping problems arising in the remote delivery of beams and pulses.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 14 June 2014
- Exact and approximate solutions for optical solitary waves in nematic
liquid crystals- Abstract: Publication date: Available online 16 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): J.M.L. MacNeil , Noel F. Smyth , Gaetano Assanto
The equations governing optical solitary waves in nonlinear nematic liquid crystals are investigated in both ( 1 + 1 ) and ( 2 + 1 ) dimensions. An isolated exact solitary wave solution is found in ( 1 + 1 ) dimensions and an isolated, exact, radially symmetric solitary wave solution is found in ( 2 + 1 ) dimensions. These exact solutions are used to elucidate what is meant by a nematic liquid crystal to have a nonlocal response and the full role of this nonlocal response in the stability of ( 2 + 1 ) dimensional solitary waves. General, approximate solitary wave solutions in ( 1 + 1 ) and ( 2 + 1 ) dimensions are found using variational methods and they are found to be in excellent agreement with full numerical solutions. These variational solutions predict that a minimum optical power is required for a solitary wave to exist in ( 2 + 1 ) dimensions, as confirmed by a careful examination of the numerical scheme and its solutions. Finally, nematic liquid crystals subjected to two different external electric fields can support the same solitary wave, exhibiting a new type of bistability.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 16 June 2014
- Regularized model of post-touchdown configurations in electrostatic MEMS:
Equilibrium analysis- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): A.E. Lindsay , J. Lega , K.B. Glasner
In canonical models of Micro-Electro Mechanical Systems (MEMS), an event called touchdown whereby the electrical components of the device come into contact, is characterized by a blow up in the governing equations and a non-physical divergence of the electric field. In the present work, we propose novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria. We employ techniques from variational calculus, dynamical systems and singular perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Crossed nonlocal effects and breakdown of the Onsager symmetry relation in
a thermodynamic description of thermoelectricity- Abstract: Publication date: Available online 16 June 2014
Source:Physica D: Nonlinear Phenomena
Author(s): A. Sellitto
Nonlocal nonlinear effects coupling the heat flux and the electric-current density in an enlarged thermodynamic description of thermoelectric systems are considered. The influence of such terms on the breakdown of the Onsager reciprocity relation between the effective transport coefficients, depending on the electric field and the temperature gradient, is analyzed up to second-order in the thermodynamic forces. The maximum value of the thermoelectric efficiency is derived as a function of the figure-of-merit and of the degree of the Onsager symmetry breaking.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: Available online 16 June 2014
- Extreme value laws for dynamical systems under observational noise
- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): Davide Faranda , Sandro Vaienti
In this paper we prove the existence of extreme value laws for dynamical systems perturbed by the instrument-like-error, also called observational noise. An orbit perturbed with observational noise mimics the behavior of an instrumentally recorded time series. Instrument characteristics–defined as precision and accuracy–act both by truncating and randomly displacing the real value of a measured observable. Here we analyze both these effects from a theoretical and a numerical point of view. First we show that classical extreme value laws can be found for orbits of dynamical systems perturbed with observational noise. Then we present numerical experiments to support the theoretical findings and give an indication of the order of magnitude of the instrumental perturbations which cause relevant deviations from the extreme value laws observed in deterministic dynamical systems. Finally, we show that the observational noise preserves the structure of the deterministic attractor. This goes against the common assumption that random transformations cause the orbits asymptotically fill the ambient space with a loss of information about the fractal structure of the attractor.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Dynamics of the wave turbulence spectrum in vibrating plates: A numerical
investigation using a conservative finite difference scheme- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): Michele Ducceschi , Olivier Cadot , Cyril Touzé , Stefan Bilbao
The dynamics of the local kinetic energy spectrum of an elastic plate vibrating in a wave turbulence (WT) regime is investigated with a finite difference, energy-conserving scheme. The numerical method allows the simulation of pointwise forcing together with realistic boundary conditions, a set-up which is close to experimental conditions. In the absence of damping, the framework of non-stationary wave turbulence is used. Numerical simulations show the presence of a front propagating to high frequencies, leaving a steady spectrum in its wake. Self-similar dynamics of the spectra are found with and without periodic external forcing. For the periodic forcing, the mean injected power is found to be constant, and the frequency at the cascade front evolves linearly with time resulting in a increase of the total energy. For the free turbulence, the energy contained in the cascade remains constant while the frequency front increases as t 1 / 3 . These self-similar solutions are found to be in accordance with the kinetic equation derived from the von Kármán plate equations. The effect of the pointwise forcing is observable and introduces a steeper slope at low frequencies, as compared to the unforced case. The presence of a realistic geometric imperfection of the plate is found to have no effect on the global properties of the spectra dynamics. The steeper slope brought by the external forcing is shown to be still observable in a more realistic case where damping is added.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Periodic orbits from second order perturbation via rational trigonometric
integrals- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): R. Prohens , J. Torregrosa
The second order Poincaré-Pontryagin-Melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers. This approach also allows us to give the shape and the period up to the first order. We address these problems for some classes of Abel differential equations and quadratic isochronous vector fields in the plane. We prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order. In particular the configurations ( 2 , 0 ) and ( 1 , 1 ) are realizable when two centers are perturbed simultaneously. The required computations show that all the considered families share the same iterated rational trigonometric integrals.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Quantification and prediction of extreme events in a one-dimensional
nonlinear dispersive wave model- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): Will Cousins , Themistoklis P. Sapsis
The aim of this work is the quantification and prediction of rare events characterized by extreme intensity in nonlinear waves with broad spectra. We consider a one-dimensional nonlinear model with deep-water waves dispersion relation, the Majda–McLaughlin–Tabak (MMT) model, in a dynamical regime that is characterized by a broadband spectrum and strong nonlinear energy transfers during the development of intermittent events with finite-lifetime. To understand the energy transfers that occur during the development of an extreme event we perform a spatially localized analysis of the energy distribution along different wavenumbers by means of the Gabor transform. A statistical analysis of the Gabor coefficients reveals (i) the low-dimensionality of the intermittent structures, (ii) the interplay between non-Gaussian statistical properties and nonlinear energy transfers between modes, as well as (iii) the critical scales (or critical Gabor coefficients) where a critical amount of energy can trigger the formation of an extreme event. We analyze the unstable character of these special localized modes directly through the system equation and show that these intermittent events are due to the interplay of the system nonlinearity, the wave dispersion, and the wave dissipation which mimics wave breaking. These localized instabilities are triggered by random localizations of energy in space, created by the dispersive propagation of low-amplitude waves with random phase. Based on these properties, we design low-dimensional functionals of these Gabor coefficients that allow for the prediction of the extreme event well before the nonlinear interactions begin to occur.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Uniqueness results for co-circular central configurations for power-law
potentials- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): Josep M. Cors , Glen R. Hall , Gareth E. Roberts
For a class of potential functions including those used for the planar n -body and n -vortex problems, we investigate co-circular central configurations whose center of mass coincides with the center of the circle containing the bodies. Useful equations are derived that completely describe the problem. Using a topological approach, it is shown that for any choice of positive masses (or circulations), if such a central configuration exists, then it is unique. It quickly follows that if the masses are all equal, then the only solution is the regular n -gon. For the planar n -vortex problem and any choice of the vorticities, we show that the only possible co-circular central configuration with center of vorticity at the center of the circle is the regular n -gon with equal vorticities.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Shift in the speed of reaction–diffusion equation with a cut-off:
Pushed and bistable fronts- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): R.D. Benguria , M.C. Depassier
We study the change in the speed of pushed and bistable fronts of the reaction–diffusion equation in the presence of a small cut-off. We give explicit formulas for the shift in the speed for arbitrary reaction terms f ( u ) . The dependence of the speed shift on the cut-off parameter is a function of the front speed and profile in the absence of the cut-off. In order to determine the speed shift we solve the leading order approximation to the front profile u ( z ) in the neighborhood of the leading edge and use a variational principle for the speed. We apply the general formula to the Nagumo equation and recover the results which have been obtained recently by geometric analysis. The formulas given are of general validity and we also apply them to a class of reaction terms which have not been considered elsewhere.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Quasiperiodicity in time evolution of the Bloch vector under the thermal
Jaynes–Cummings model- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
Author(s): Hiroo Azuma , Masashi Ban
We study a quasiperiodic structure in the time evolution of the Bloch vector, whose dynamics is governed by the thermal Jaynes–Cummings model (JCM). Putting the two-level atom into a certain pure state and the cavity field into a mixed state in thermal equilibrium at initial time, we let the whole system evolve according to the JCM Hamiltonian. During this time evolution, motion of the Bloch vector seems to be in disorder. Because of the thermal photon distribution, both a norm and a direction of the Bloch vector change hard at random. In this paper, taking a different viewpoint compared with ones that we have been used to, we investigate quasiperiodicity of the Bloch vector’s trajectories. Introducing the concept of the quasiperiodic motion, we can explain the confused behaviour of the system as an intermediate state between periodic and chaotic motions. More specifically, we discuss the following two facts: (1) If we adjust the time interval Δ t properly, figures consisting of plotted dots at the constant time interval acquire scale invariance under replacement of Δ t by s Δ t , where s ( > 1 ) is an arbitrary real but not transcendental number. (2) We can compute values of the time variable t , which let S z ( t ) (the absolute value of the z -component of the Bloch vector) be very small, with the Diophantine approximation (a rational approximation of an irrational number).
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- Editorial Board
- Abstract: Publication date: 1 July 2014
Source:Physica D: Nonlinear Phenomena, Volumes 280–281
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 1 July 2014
- An ODE for boundary layer separation on a sphere and a hyperbolic space
- Abstract: Publication date: 15 July 2014
Source:Physica D: Nonlinear Phenomena, Volume 282
Author(s): Chi Hin Chan , Magdalena Czubak , Tsuyoshi Yoneda
Ma and Wang derived an equation linking the separation location and times for the boundary layer separation of incompressible fluid flows. The equation gave a necessary condition for the separation (bifurcation) point. The purpose of this paper is to generalize the equation to other geometries, and to phrase it as a simple ODE. Moreover we consider the Navier–Stokes equation with the Coriolis effect, which is related to the presence of trade winds on Earth.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 15 July 2014
- Role of non-ideality for the ion transport in porous media: Derivation of
the macroscopic equations using upscaling- Abstract: Publication date: 15 July 2014
Source:Physica D: Nonlinear Phenomena, Volume 282
Author(s): Grégoire Allaire , Robert Brizzi , Jean-François Dufrêche , Andro Mikelić , Andrey Piatnitski
This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N -component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of the Poisson–Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for the Poisson–Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O’Brien’s argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 15 July 2014
- Aperiodic dynamics in a deterministic adaptive network model of attitude
formation in social groups- Abstract: Publication date: 15 July 2014
Source:Physica D: Nonlinear Phenomena, Volume 282
Author(s): Jonathan A. Ward , Peter Grindrod
Adaptive network models, in which node states and network topology coevolve, arise naturally in models of social dynamics that incorporate homophily and social influence. Homophily relates the similarity between pairs of nodes’ states to their network coupling strength, whilst social influence causes coupled nodes’ states to convergence. In this paper we propose a deterministic adaptive network model of attitude formation in social groups that includes these effects, and in which the attitudinal dynamics are represented by an activator–inhibitor process. We illustrate that consensus, corresponding to all nodes adopting the same attitudinal state and being fully connected, may destabilise via Turing instability, giving rise to aperiodic dynamics with sensitive dependence on initial conditions. These aperiodic dynamics correspond to the formation and dissolution of sub-groups that adopt contrasting attitudes. We discuss our findings in the context of cultural polarisation phenomena.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 15 July 2014
- Discrete set of kink velocities in Josephson structures: The nonlocal
double sine–Gordon model- Abstract: Publication date: 15 July 2014
Source:Physica D: Nonlinear Phenomena, Volume 282
Author(s): G.L. Alfimov , A.S. Malishevskii , E.V. Medvedeva
We study a model of Josephson layered structure which is characterized by two peculiarities: (i) superconducting layers are thin; (ii) the current–phase relation is non-sinusoidal and is described by two sine harmonics. The governing equation is a nonlocal generalization of double sine–Gordon (NDSG) equation. We argue that the dynamics of fluxons in the NDSG model is unusual. Specifically, we show that there exists a set of particular constant velocities (called “sliding” velocities) for non-radiating stationary fluxon propagation. In dynamics, the presence of this set results in quantization of fluxon velocities: in numerical experiments a traveling kink-like excitation radiates energy and slows down to one of these particular constant velocities, taking the shape of predicted 2 π -kink. We conjecture that the set of these stationary velocities is infinite and present an asymptotic formula for them.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 15 July 2014
- Editorial Board
- Abstract: Publication date: 15 July 2014
Source:Physica D: Nonlinear Phenomena, Volume 282
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 15 July 2014
- Periodic solutions of gene networks with steep sigmoidal regulatory
functions- Abstract: Publication date: 15 July 2014
Source:Physica D: Nonlinear Phenomena, Volume 282
Author(s): Roderick Edwards , Liliana Ironi
We address the question of existence and stability of periodic solutions in gene regulatory networks. The threshold-dependent network dynamics divides the phase space into domains and a qualitative description can be derived, specifying which transitions between domains can occur. Any periodic solution must follow a cyclic sequence of domains, but the problem of determining when such a cyclic sequence of domains contains a periodic solution, and when it is stable, has not been completely resolved, though results have been obtained before for restricted classes of networks. Here, we develop a method by which existence or non-existence of such solutions can be demonstrated analytically in any given example of a general class of gene networks with steep sigmoidal interactions, under the assumption that any gene product that regulates multiple other genes does so at distinct thresholds. Our method determines qualitative stability, but we also give a procedure that, where applicable, allows determination of quantitative stability of a periodic solution. This complements the previous development of a local analysis method for this class of systems, which allows computation of trajectories through any sequence of domains. Together the previous and current work form the basis for rigorous computer-aided assessment of qualitative dynamics of a very general class of gene network models. The ability to handle periodic solutions will also increase the applicability of such a computational tool to the design of synthetic networks.
PubDate: 2014-06-18T15:10:07Z
- Abstract: Publication date: 15 July 2014
- Shearless transport barriers in unsteady two-dimensional flows and maps
- Abstract: Publication date: Available online 13 April 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Mohammad Farazmand , Daniel Blazevski , George Haller
We develop a variational principle that extends the notion of a shearless transport barrier from steady to general unsteady two-dimensional flows and maps defined over a finite time interval. This principle reveals that hyperbolic Lagrangian Coherent Structures (LCSs) and parabolic LCSs (or jet cores) are the two main types of shearless barriers in unsteady flows. Based on the boundary conditions they satisfy, parabolic barriers are found to be more observable and robust than hyperbolic barriers, confirming widespread numerical observations. Both types of barriers are special null-geodesics of an appropriate Lorentzian metric derived from the Cauchy–Green strain tensor. Using this fact, we devise an algorithm for the automated computation of parabolic barriers. We illustrate our detection method on steady and unsteady non-twist maps and on the aperiodically forced Bickley jet.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: Available online 13 April 2014
- Orientational dynamics of weakly inertial axisymmetric particles in steady
viscous flows- Abstract: Publication date: Available online 16 April 2014
Source:Physica D: Nonlinear Phenomena
Author(s): J. Einarsson , J.R. Angilella , B. Mehlig
The orientational dynamics of weakly inertial axisymmetric particles in a steady flow is investigated. We derive an asymptotic equation of motion for the unit axial vector along the particle symmetry axis, valid for small Stokes number St , and for any axisymmetric particle in any steady linear viscous flow. This reduced dynamics is analysed in two ways, both pertain to the case of a simple shear flow. In this case inertia induces a coupling between precession and nutation. This coupling affects the dynamics of the particle, breaks the degeneracy of the Jeffery orbits, and creates two limiting periodic orbits. We calculate the leading-order Floquet exponents of the limiting periodic orbits and show analytically that prolate objects tend to a tumbling orbit, while oblate objects tend to a log-rolling orbit, in agreement with previous analytical and numerical results. Second, we analyse the role of the limiting orbits when rotational noise is present. We formulate the Fokker–Planck equation describing the orientational distribution of an axisymmetric particle, valid for small St and general Péclet number Pe . Numerical solutions of the Fokker–Planck equation, obtained by means of expansion in spherical harmonics, show that stationary orientational distributions are close to the inertia-free case when Pe St ≪ 1 , whereas they are determined by inertial effects, though small, when Pe ≫ 1 / St ≫ 1 .
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: Available online 16 April 2014
- Inverse scattering transform for 3-level coupled Maxwell–Bloch
equations with inhomogeneous broadening- Abstract: Publication date: Available online 18 April 2014
Source:Physica D: Nonlinear Phenomena
Author(s): S. Chakravarty , B. Prinari , M.J. Ablowitz
In this paper we study the propagation of optical pulses in an optical medium with coherent three-level atomic transitions. The interaction between the pulses and the medium is described by the coupled Maxwell–Bloch equations, which we investigate by applying the method of inverse scattering transform. The details of the inverse scattering method and the non-trivial evolution of the associated scattering data are discussed. The one- and two-soliton solutions, polarization shifts due to two-soliton interactions, and the explicit form of the transmission matrix associated with pure soliton solutions are also derived.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: Available online 18 April 2014
- Solution of the Fokker–Planck equation in a wind turbine array
boundary layer- Abstract: Publication date: Available online 19 April 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Matthew.S. Melius , Murat Tutkun , Raúl Bayoán Cal
Hot-wire velocity signals from a model wind turbine array boundary layer flow wind tunnel experiment are analyzed. In confirming Markovian properties, a description of the evolution of the probability density function of velocity increments via a Fokker–Planck equation is attained. A Fokker–Planck equation is possible due to the direct computation of the drift and diffusion coefficients from the experimental measurement data which were acquired within the turbine canopy. A good agreement is observed in the probability density functions between the experimental data and numerical solutions resulting from the Fokker–Planck equation, especially in the far-wake region. The results serve as a tool for improved estimation of wind velocity within the array and provide evidence that the evolution of such complex and turbulent flow is also governed by a Fokker–Planck equation at certain scales.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: Available online 19 April 2014
- Bifurcation analysis of a model for atherosclerotic plaque evolution
- Abstract: Publication date: Available online 21 April 2014
Source:Physica D: Nonlinear Phenomena
Author(s): M.A.K. Bulelzai , J.L.A. Dubbeldam , H.G.E. Meijer
We analyze two ordinary differential equation (ODE) models for atherosclerosis. The ODE models describe long time evolution of plaques in arteries. We show how the dynamics of the first atherosclerosis model (model A) can be understood using codimension-two bifurcation analysis. The Low-Density Lipoprotein (LDL) intake parameter ( d ) is the first control parameter and the second control parameter is either taken to be the conversion rate of macrophages ( b ) or the wall shear stress ( σ ). Our analysis reveals that in both cases a Bogdanov-Takens (BT) point acts as an organizing center. The bifurcation diagrams are calculated partly analytically and to a large extent numerically using AUTO07 and MATCONT. The bifurcation curves show that the concentration of LDL in the plaque as well as the monocyte and the macrophage concentration exhibit oscillations for a certain range of values of the control parameters. Moreover, we find that there are threshold values for both the cholesterol intake rate d c r i t and the conversion rate of the macrophages b c r i t , which depend on the values of other parameters, above which the plaque volume increases with time. It is found that larger conversion rates of macrophages lower the threshold value of cholesterol intake and vice versa. We further argue that the dynamics for model A can still be discerned in the second model (model B) in which the slow evolution of the radius of the artery is coupled self-consistently to changes in the plaque volume. The very slow evolution of the radius of the artery compared to the other processes makes it possible to use a slow manifold approximation to study the dynamics in this case. We find that in this case the model predicts that the concentrations of the plaque constituents may go through a period of oscillations before the radius of the artery will start to decrease. These oscillations hence act as a precursor for the reduction of the artery radius by plaque growth.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: Available online 21 April 2014
- Painlevé IV: A numerical study of the fundamental domain and beyond
- Abstract: Publication date: Available online 24 April 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Jonah A. Reeger , Bengt Fornberg
The six Painlevé equations were introduced over a century ago, motivated by rather theoretical considerations. Over the last several decades, these equations and their solutions, known as the Painlevé transcendents, have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast ‘pole field solver’ (Fornberg and Weideman (2011)). The fourth Painlevé equation has two free parameters in its coefficients, as well as two free initial conditions. After summarizing key analytical results for P IV , the present study applies this new computational tool to the fundamental domain and a surrounding region of the parameter space. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: Available online 24 April 2014
- Exponential synchronization of Kuramoto oscillators using spatially local
coupling- Abstract: Publication date: 1 June 2014
Source:Physica D: Nonlinear Phenomena, Volume 277
Author(s): Jong-Ho Kim , Jea-Hyun Park
We study the generalized Kuramoto model of coupled phase oscillators with a finite size, and discuss the asymptotic complete phase–frequency synchronization. The generalized Kuramoto model has inherent difficulties in mathematical approaches that this model is governed by nonlinear equations and the Kuramoto oscillator is arbitrarily connected with the others. To overcome these mathematical barriers, many researchers have adopted a linearization of homogeneous solutions, and applied a perturbation method. However, we introduce a new method which just requires some conditions on the smallest and largest positive eigenvalues of the graph Laplacian, and directly compute the bounds of homogeneous solutions. Using this method, we present analytic results for the generalized Kuramoto model. More specifically, we give a few sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our sufficient conditions and decay rate depend on the coupling strength, the initial phase and natural frequency configurations, and the graph Laplacian, but the conditions are independent of the system size. Moreover, we estimate the time evolution of deviations for the phase and frequency, and show that the smallest and largest positive eigenvalues for the graph Laplacian affect the stability region and convergence rate for the synchronized states. Finally, we compare our analytic results with numerical simulations using a few examples.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: 1 June 2014
- Spatio-temporal oscillations in the Keller–Segel system with
logistic growth- Abstract: Publication date: 1 June 2014
Source:Physica D: Nonlinear Phenomena, Volume 277
Author(s): Shin-Ichiro Ei , Hirofumi Izuhara , Masayasu Mimura
The Keller–Segel system with the logistic growth term is discussed from the spatio-temporal-oscillation point of view. This system exhibits two different types of spatio-temporal oscillations in certain distinct parameter regimes. In this paper, we study the difference between the two types of spatio-temporal oscillations. In particular, the characteristic properties of the behaviors become clear in a limiting system when a certain parameter value tends to zero. Moreover, we demonstrate that the onset of one of the spatio-temporal oscillatory patterns is an infinite-dimensional relaxation oscillation that consists of slow and fast dynamics.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: 1 June 2014
- Networks of diffusively time-delay coupled systems: Conditions for
synchronization and its relation to the network topology- Abstract: Publication date: 1 June 2014
Source:Physica D: Nonlinear Phenomena, Volume 277
Author(s): Erik Steur , Wim Michiels , Henri Huijberts , Henk Nijmeijer
We consider networks of time-delayed diffusively coupled systems and relate conditions for synchronization of the systems in the network to the topology of the network. First we present sufficient conditions for the solutions of the time-delayed coupled systems to be bounded. Next we give conditions for local synchronization and we show that the values of the coupling strength and time-delay for which there is local synchronization in any network can be determined from these conditions. In addition we present results on global synchronization in relation to the network topology for networks of a class of nonlinear systems. We illustrate our results with examples of synchronization in networks with FitzHugh–Nagumo model neurons and Hindmarsh–Rose neurons.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: 1 June 2014
- Editorial Board
- Abstract: Publication date: 1 June 2014
Source:Physica D: Nonlinear Phenomena, Volume 277
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: 1 June 2014
- The effect of boundaries on the asymptotic wavenumber of spiral wave
solutions of the complex Ginzburg–Landau equation- Abstract: Publication date: 15 June 2014
Source:Physica D: Nonlinear Phenomena, Volumes 278–279
Author(s): M. Aguareles
In this paper we consider an oscillatory medium whose dynamics are modeled by the complex Ginzburg–Landau equation. In particular, we focus on n -armed spiral wave solutions of the complex Ginzburg–Landau equation in a disk of radius d with homogeneous Neumann boundary conditions. It is well-known that such solutions exist for small enough values of the twist parameter q and large enough values of d . We investigate the effect of boundaries on the rotational frequency of the spirals, which is an unknown of the problem uniquely determined by the parameters d and q . We show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak. We use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q .
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: 15 June 2014
- Modeling selective local interactions with memory: Motion on a 2D lattice
- Abstract: Publication date: 15 June 2014
Source:Physica D: Nonlinear Phenomena, Volumes 278–279
Author(s): Daniel Weinberg , Doron Levy
We consider a system of particles that simultaneously move on a two-dimensional periodic lattice at discrete times steps. Particles remember their last direction of movement and may either choose to continue moving in this direction, remain stationary, or move toward one of their neighbors. The form of motion is chosen based on predetermined stationary probabilities. Simulations of this model reveal a connection between these probabilities and the emerging patterns and size of aggregates. In addition, we develop a reaction–diffusion master equation from which we derive a system of ODEs describing the dynamics of the particles on the lattice. Simulations demonstrate that solutions of the ODEs may replicate the aggregation patterns produced by the stochastic particle model. We investigate conditions on the parameters that influence the locations at which particles prefer to aggregate. This work is a two-dimensional generalization of Galante and Levy (2012), in which the corresponding one-dimensional problem was studied.
PubDate: 2014-04-29T06:46:58Z
- Abstract: Publication date: 15 June 2014